Rational Points on K3 Surfaces and Derived Equivalence

Abstract

The geometry of vector bundles and derived categories on complex K3 surfaces has developed rapidly since Mukai's seminal work [Muk87]. Many foundational questions have been answered:• the existence of vector bundles and twisted sheaves with prescribed invariants; , and other authors. Our focus in this note is on rational points over non-closed fields of arithmetic interest. We seek to relate the notion of derived equivalence to arithmetic problems over various fields. Our guiding questions are: Question 1. Let X and Y be K3 surfaces, derived equivalent over a field F . Does the existence/density of rational points of X imply the same for Y ? Given α ∈ Br(X), let (X, α) denote the twisted K3 surface associated with α, i.e., if P → X is an étale projective bundle representing α, of relative dimension r − 1 then (X, α) = [P/ SL r ].Question 2. Suppose that (X, α) and (Y, β) are derived equivalent over F . Does the existence of a rational point on the former imply the same for the latter?Date: September 9, 2015. Note that an F -rational point of (X, α) corresponds to an x ∈ X(F ) such that α|x = 0 ∈ Br(F ). After this paper was released, Ascher, Dasaratha, Perry, and Zhou [ADPZ15] found that Question 2 has a negative answer, even over local fields.We shall consider these questions for F finite, p-adic, real, and local with algebraically closed residue field. These will serve as a foundation for studying how the geometry of K3 surfaces interacts with Diophantine questions over local and global fields. For instance, is the Hasse/Brauer-Manin formalism over global fields compatible with (twisted) derived equivalence? See [HVAV11, HVA13, MSTVA14] for concrete applications to rational points problems.In this paper, we first review general properties of derived equivalence over arbitrary base fields. We then offer examples which illuminate some of the challenges in applying derived category techniques. The case of finite and real fields is presented first-here the picture is well developed. Local fields of equicharacteristic zero are also fairly well understood, at least for K3 surfaces with semistable or other mild reduction. The analogous questions in mixed characteristic remain largely open, but comparison with the geometric case suggests a number of avenues for future investigation. Acknowledgments:We are grateful to Jean-Louis Colliot-Thélène,

Semiorthogonal decompositions and birational geometry of del Pezzo surfaces over arbitrary fields

“…For del Pezzo surfaces of Picard rank 1 and degree at least 5, a consequence of our results is that the index of can be calculated only in terms of the second Chern classes of generators of the three blocks, improving upon, in this case, a result of Hassett and Tschinkel [, Lemma 8]. Theorem Let be a del Pezzo surface of degree .…”

“…This question, which was the original motivation for the current work, is now central to a growing body of research into arithmetic aspects of the theory of derived categories, see . As an example, if is a smooth projective surface, Hassett and Tschinkel [, Lemma 8] prove that the index of can be recovered from as the greatest common divisor of the second Chern classes of objects. Recall that the index of a variety over is the greatest common divisor of the degrees of closed points of .…”

Good reduction of K3 surfaces

“…In [HT17, Theorem 35], a similar result is obtained for K3 surfaces over , but their proof uses methods different from ours. As in the case of Abelian varieties in [ST68], we obtain the following independence of .…”

Rational Points on Twisted K3 Surfaces and Derived Equivalences

“…A twisted K3 surface is a pair (X, α), where X is a K3 surface and α ∈ Br(X) is a Brauer class. In a recent survey paper [HT14], Hassett and Tschinkel asked whether the existence of a rational point on a twisted K3 surface is invariant under derived equivalence. More precisely, they asked:…”

“…In [HT14], it is shown that for the untwisted case of the question where α 1 , α 2 vanish, the answer is positive over certain fields k, e.g. R, finite fields, and p-adic fields (provided the X i have good reduction, or p ≥ 7 and the X i have ADE reduction).…”